Learning t-doped stabilizer states

Quantum 8, 1361 (2024).

https://doi.org/10.22331/q-2024-05-27-1361

In this paper, we present a learning algorithm aimed at learning states obtained from computational basis states by Clifford circuits doped with a finite number $t$ of $T$-gates. The algorithm learns an exact tomographic description of $t$-doped stabilizer states in terms of Pauli observables. This is possible because such states are countable and form a discrete set. To tackle the problem, we introduce a novel algebraic framework for $t$-doped stabilizer states, which extends beyond $T$-gates and includes doping with any kind of local non-Clifford gate. The algorithm requires resources of complexity $operatorname{poly}(n,2^t)$ and exhibits an exponentially small probability of failure.